Theorie de la fonctionnnelle de la densite

avec separation de porteepour les forces de van der Waals

Julien Toulouse1

Iann Gerber2, Georg Jansen3, Andreas Savin1, Janos Angyan4

1 Laboratoire de Chimie Theorique, UPMC Univ Paris 06 et CNRS, Paris, France

2 Universite de Toulouse, INSA-UPS, LPCNO, Toulouse, France3 Fachbereich Chemie, Universitat Duisburg-Essen, Essen, Germany

4 CRM2, Institut Jean Barriol, Universite de Nancy et CNRS, Vandoeuvre-les-Nancy,

France

Email : julien.toulouse@upmc.fr

Page web : www.lct.jussieu.fr/pagesperso/toulouse/

novembre 2008

1 Kohn-Sham DFT and ACFDT approaches

2 Range-separated multideterminant DFT

3 Short-range density functionals

4 Range-separated ACFDT method

5 Some results

1 Kohn-Sham DFT and ACFDT approaches

2 Range-separated multideterminant DFT

3 Short-range density functionals

4 Range-separated ACFDT method

5 Some results

Kohn-Sham DFT

Kohn-Sham (KS) scheme

E = minΦ

{

〈Φ|T + Vne |Φ〉 + EH[nΦ] + Exc [nΦ]}

Φ : single-determinant wave function

Kohn-Sham DFT

Kohn-Sham (KS) scheme

E = minΦ

{

〈Φ|T + Vne |Φ〉 + EH[nΦ] + Exc [nΦ]}

Φ : single-determinant wave function

One problem (among others):Usual approximations for exchange-correlation functional Exc [n](LDA, GGA, ...) do not describe well (long-range) van derWaals dispersion forces

Example: interaction energy curve of Ne2

LDA and PBE functionals, aug-cc-pV5Z basis:

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

5 6 7 8 9 10

Inte

ract

ion

en

ergy (

mH

art

ree)

Interatomic distance (Bohr)

AccurateLDAPBE

Ne2

ACFDT approach to DFT

Starting from the adiabatic connection formula for correlation energy:

Ec =

∫ 1

0

dλ{

〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}

ACFDT approach to DFT

Starting from the adiabatic connection formula for correlation energy:

Ec =

∫ 1

0

dλ{

〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}

or, with a compact notation,

Ec =1

2

∫ 1

0

dλ Tr [wee ∗ Pc,λ]

ACFDT approach to DFT

Starting from the adiabatic connection formula for correlation energy:

Ec =

∫ 1

0

dλ{

〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}

or, with a compact notation,

Ec =1

2

∫ 1

0

dλ Tr [wee ∗ Pc,λ]

and using the fluctuation-dissipation theorem

Pc,λ = −1

π

∫

∞

0

dω [χλ(iω) − χKS(iω)]

ACFDT approach to DFT

Starting from the adiabatic connection formula for correlation energy:

Ec =

∫ 1

0

dλ{

〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}

or, with a compact notation,

Ec =1

2

∫ 1

0

dλ Tr [wee ∗ Pc,λ]

and using the fluctuation-dissipation theorem

Pc,λ = −1

π

∫

∞

0

dω [χλ(iω) − χKS(iω)]

leads to

Ec = −1

2π

∫ 1

0

dλ

∫

∞

0

dω Tr [wee ∗ (χλ(iω) − χKS(iω))]

ACFDT approach to DFT

Starting from the adiabatic connection formula for correlation energy:

Ec =

∫ 1

0

dλ{

〈Ψλ|Wee |Ψλ〉 − 〈ΦKS|Wee |ΦKS〉}

or, with a compact notation,

Ec =1

2

∫ 1

0

dλ Tr [wee ∗ Pc,λ]

and using the fluctuation-dissipation theorem

Pc,λ = −1

π

∫

∞

0

dω [χλ(iω) − χKS(iω)]

leads to

Ec = −1

2π

∫ 1

0

dλ

∫

∞

0

dω Tr [wee ∗ (χλ(iω) − χKS(iω))]

where the response function χλ(iω) is given by

χλ(iω)−1 = χKS(iω)−1 − fHxc,λ(iω)

Random Phase Approximation (RPA)

RPA approximation: fxc,λ = 0

Random Phase Approximation (RPA)

RPA approximation: fxc,λ = 0

So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA

Random Phase Approximation (RPA)

RPA approximation: fxc,λ = 0

So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA

=⇒ increasing interest in the DFT community:Perdew, Dobson, Furche, Gonze, Kresse, Scuseria, ...

Random Phase Approximation (RPA)

RPA approximation: fxc,λ = 0

So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA

=⇒ increasing interest in the DFT community:Perdew, Dobson, Furche, Gonze, Kresse, Scuseria, ...

Encouraging results:

consistent with exact exchange

correct dispersion forces at (very) large separation

good cohesive energies and lattice constants of solids

some improvement in description of bond dissociation

Random Phase Approximation (RPA)

RPA approximation: fxc,λ = 0

So, total energy is E = 〈ΦKS|T + Vne |ΦKS〉 + EH + Ex ,exact + Ec,RPA

=⇒ increasing interest in the DFT community:Perdew, Dobson, Furche, Gonze, Kresse, Scuseria, ...

Encouraging results:

consistent with exact exchange

correct dispersion forces at (very) large separation

good cohesive energies and lattice constants of solids

some improvement in description of bond dissociation

But several unsatisfactory aspects:

correlation energies far too negative

strong dependence on basis size

bump at intermediate distances in some dissociation curves

dependence on input orbitals

embarrassing results for simple van der Waals dimers!

Example: interaction energy curve of Ne2

RPA (with PBE orbitals), aug-cc-pV5Z basis:

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

5 6 7 8 9 10

Inte

ract

ion

en

ergy (

mH

art

ree)

Interatomic distance (Bohr)

AccurateRPA

Ne2

Example: interaction energy curve of Be2

RPA (with PBE orbitals), cc-pV5Z basis:

-4

-2

0

2

4

4 5 6 7 8 9 10 11

Inte

ract

ion

en

ergy (

mH

art

ree)

Interatomic distance (Bohr)

AccurateRPA

Be2

1 Kohn-Sham DFT and ACFDT approaches

2 Range-separated multideterminant DFT

3 Short-range density functionals

4 Range-separated ACFDT method

5 Some results

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

W lree =

∑

i<j

erf(µrij)

rij: long-range electron-electron interaction

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

W lree =

∑

i<j

erf(µrij)

rij: long-range electron-electron interaction

E srHxc [n] : short-range Hxc density functional

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

W lree =

∑

i<j

erf(µrij)

rij: long-range electron-electron interaction

E srHxc [n] : short-range Hxc density functional

minimizing wave function Ψlr

=∑∑∑

i ciΦi is multi-determinant

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

W lree =

∑

i<j

erf(µrij)

rij: long-range electron-electron interaction

E srHxc [n] : short-range Hxc density functional

minimizing wave function Ψlr

=∑∑∑

i ciΦi is multi-determinant

parameter µ controls the range of separation.Limiting cases:µ = 0 =⇒ KS DFTµ → ∞ =⇒ Standard wave function methods

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

W lree =

∑

i<j

erf(µrij)

rij: long-range electron-electron interaction

E srHxc [n] : short-range Hxc density functional

minimizing wave function Ψlr

=∑∑∑

i ciΦi is multi-determinant

parameter µ controls the range of separation.Limiting cases:µ = 0 =⇒ KS DFTµ → ∞ =⇒ Standard wave function methods

In principle: exact

Range-separated multideterminant DFT

Multideterminant extension of KS scheme with range separation

Ground-state energy:

E = minΨ

{

〈Ψ|T + Vne + W lree |Ψ〉 + E sr

Hxc [nΨ]}

W lree =

∑

i<j

erf(µrij)

rij: long-range electron-electron interaction

E srHxc [n] : short-range Hxc density functional

minimizing wave function Ψlr

=∑∑∑

i ciΦi is multi-determinant

parameter µ controls the range of separation.Limiting cases:µ = 0 =⇒ KS DFTµ → ∞ =⇒ Standard wave function methods

In principle: exact

In practice: approximations are necessary for Ψlr and E sr

xc [n]

Range-separated multideterminant DFT: approximations

Approximations for E srxc [n]

short-range LDA

short-range GEA

short-range GGA

...

Range-separated multideterminant DFT: approximations

Approximations for E srxc [n]

short-range LDA

short-range GEA

short-range GGA

...

Approximations for Ψlr

single-determinant =⇒ HF+DFT method (or RSH method)

MCSCF =⇒ MCSCF+DFT method (for near-degeneracy)

CI =⇒ CI+DFT method

CC =⇒ CC+DFT method

MP2 =⇒ RSH+MP2 method (for van der Waals)

RPA or TDHF =⇒ RSH+TDHF method (for van der Waals)

...

1 Kohn-Sham DFT and ACFDT approaches

2 Range-separated multideterminant DFT

3 Short-range density functionals

4 Range-separated ACFDT method

5 Some results

Short-range exchange energy: LDA

Esr,µx ,LDA[n] =

∫

n(r) εsr,µx ,unif(n(r))dr

Short-range exchange energy: LDA

Esr,µx ,LDA[n] =

∫

n(r) εsr,µx ,unif(n(r))dr

For Be atom:

0 2 4 6 8Μ Ha.u.L

-2.5

-2

-1.5

-1

-0.5

0

Exsr,Μ

Ha.u.L

exactLDA

LDA accuratefor a short-range interaction

Short-range exchange energy: LDA

Esr,µx ,LDA[n] =

∫

n(r) εsr,µx ,unif(n(r))dr

For Be atom:

0 2 4 6 8Μ Ha.u.L

-2.5

-2

-1.5

-1

-0.5

0

Exsr,Μ

Ha.u.L

exactLDA

LDA accuratefor a short-range interaction

Asymptotic expansion for µ → ∞ :

E sr,µx = −

A1

µ2

∫

n(r)2dr +A2

µ4

∫

n(r)

(

|∇n(r)|2

2n(r)+ 4τ(r)

)

dr + · · ·

Short-range correlation energy: LDA

Esr,µc,LDA[n] =

∫

n(r) εsr,µc,unif(n(r))dr

Short-range correlation energy: LDA

Esr,µc,LDA[n] =

∫

n(r) εsr,µc,unif(n(r))dr

For Be atom:

0 2 4 6 8Μ Ha.u.L

-0.2

-0.15

-0.1

-0.05

0

Ecsr,Μ

Ha.u.L

exactLDA

LDA accuratefor a short-range interaction

Short-range correlation energy: LDA

Esr,µc,LDA[n] =

∫

n(r) εsr,µc,unif(n(r))dr

For Be atom:

0 2 4 6 8Μ Ha.u.L

-0.2

-0.15

-0.1

-0.05

0

Ecsr,Μ

Ha.u.L

exactLDA

LDA accuratefor a short-range interaction

Asymptotic expansion for µ → ∞ :

E sr,µc =

B1

µ2

∫

n2,c(r, r)dr +B2

µ3

∫

n2(r, r)dr + · · ·

Short-range exchange energy: GGA

Short-range GGA functional of Heyd, Scuseria and Ernzerhof(2003) based on the PBE exchange hole:

εsr,µx ,GGA(n) =

1

2

∫

nx ,PBE(n, |∇n|, r12)w sr,µee (r12)dr12

For Be atom:

0 1 2 3 4 5 6Μ Ha.u.L

-2.5

-2

-1.5

-1

-0.5

0

Exsr,Μ

Ha.u.L

exactLDAGGA

=⇒ GGA describes well a longer range of interaction

Short-range correlation energy: GGA

Interpolation between PBE at µ = 0 and expansion of LDA for µ → ∞:

εsr,µc,GGA(n, |∇n|) =

εc,PBE(n, |∇n|)

1 + d1(n)µ + d2(n)µ2

For Be atom:

0 1 2 3 4 5 6Μ Ha.u.L

-0.2

-0.15

-0.1

-0.05

0

Ecsr,Μ

Ha.u.LexactLDAGGA

=⇒ GGA describes well a longer range of interaction

1 Kohn-Sham DFT and ACFDT approaches

2 Range-separated multideterminant DFT

3 Short-range density functionals

4 Range-separated ACFDT method

5 Some results

Range-separated hybrid (RSH) scheme

Restriction to single-determinant wave functions Φ:

ERSH = minΦ

{

〈Φ|T + Vne + W lree |Φ〉 + E sr

Hxc [nΦ]}

Range-separated hybrid (RSH) scheme

Restriction to single-determinant wave functions Φ:

ERSH = minΦ

{

〈Φ|T + Vne + W lree |Φ〉 + E sr

Hxc [nΦ]}

The minimizing RSH determinant ΦRSH is given by

(

T + Vne + V lrHx ,HF + V sr

Hxc

)

|ΦRSH〉 = E0|ΦRSH〉,

Range-separated hybrid (RSH) scheme

Restriction to single-determinant wave functions Φ:

ERSH = minΦ

{

〈Φ|T + Vne + W lree |Φ〉 + E sr

Hxc [nΦ]}

The minimizing RSH determinant ΦRSH is given by

(

T + Vne + V lrHx ,HF + V sr

Hxc

)

|ΦRSH〉 = E0|ΦRSH〉,

So the RSH energy is

ERSH = 〈ΦRSH|T+Vne |ΦRSH〉+EH[nΦRSH]+E lr

x ,HF[ΦRSH]+E srxc [nΦRSH

]

Adiabatic connection starting from RSH

Exact energy = RSH energy + long-range correlation energy

E = ERSH + E lrc

Adiabatic connection starting from RSH

Exact energy = RSH energy + long-range correlation energy

E = ERSH + E lrc

Let’s define the following adiabatic connection

Eλ = minΨ

{

〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr

Hxc [nΨ]}

with the long-range perturbation operator

W lr = W lree − V lr

Hx ,HF

Adiabatic connection starting from RSH

Exact energy = RSH energy + long-range correlation energy

E = ERSH + E lrc

Let’s define the following adiabatic connection

Eλ = minΨ

{

〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr

Hxc [nΨ]}

with the long-range perturbation operator

W lr = W lree − V lr

Hx ,HF

minimizing wave function Ψlrλ is multideterminant

Adiabatic connection starting from RSH

Exact energy = RSH energy + long-range correlation energy

E = ERSH + E lrc

Let’s define the following adiabatic connection

Eλ = minΨ

{

〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr

Hxc [nΨ]}

with the long-range perturbation operator

W lr = W lree − V lr

Hx ,HF

minimizing wave function Ψlrλ is multideterminant

Limits:For λ = 0: Ψ

lrλ=0 = ΦRSH

For λ = 1: Ψlrλ=1 = Ψ

lr and Eλ=1 = E

Adiabatic connection starting from RSH

Exact energy = RSH energy + long-range correlation energy

E = ERSH + E lrc

Let’s define the following adiabatic connection

Eλ = minΨ

{

〈Ψ|T + Vne + V lrHx ,HF + λW lr|Ψ〉 + E sr

Hxc [nΨ]}

with the long-range perturbation operator

W lr = W lree − V lr

Hx ,HF

minimizing wave function Ψlrλ is multideterminant

Limits:For λ = 0: Ψ

lrλ=0 = ΦRSH

For λ = 1: Ψlrλ=1 = Ψ

lr and Eλ=1 = E

the density is NOT constant on the adiabatic connection

Long-range correlation energy Elrc

We have the following adiabatic connection formula:

E lrc =

∫ 1

0

dλ{

〈Ψlrλ|W

lr|Ψlrλ〉 − 〈ΦRSH|W

lr|ΦRSH〉}

Long-range correlation energy Elrc

We have the following adiabatic connection formula:

E lrc =

∫ 1

0

dλ{

〈Ψlrλ|W

lr|Ψlrλ〉 − 〈ΦRSH|W

lr|ΦRSH〉}

=1

2

∫ 1

0

dλ Tr[

w lr ∗ P lrc,λ

]

Long-range correlation energy Elrc

We have the following adiabatic connection formula:

E lrc =

∫ 1

0

dλ{

〈Ψlrλ|W

lr|Ψlrλ〉 − 〈ΦRSH|W

lr|ΦRSH〉}

=1

2

∫ 1

0

dλ Tr[

w lr ∗ P lrc,λ

]

and using the fluctuation-dissipation theorem

P lrc,λ = −

1

π

∫

∞

0

dω[

χlrλ(iω) − χRSH(iω)

]

+ ∆lrλ

where ∆lrλ comes from the variation of the density. So

Long-range correlation energy Elrc

We have the following adiabatic connection formula:

E lrc =

∫ 1

0

dλ{

〈Ψlrλ|W

lr|Ψlrλ〉 − 〈ΦRSH|W

lr|ΦRSH〉}

=1

2

∫ 1

0

dλ Tr[

w lr ∗ P lrc,λ

]

and using the fluctuation-dissipation theorem

P lrc,λ = −

1

π

∫

∞

0

dω[

χlrλ(iω) − χRSH(iω)

]

+ ∆lrλ

where ∆lrλ comes from the variation of the density. So

E lrc = −

1

2π

∫ 1

0

dλ

∫

∞

0

dω Tr[

w lr ∗(

χlrλ(iω) − χRSH(iω)

)]

+1

2

∫ 1

0

dλ Tr[

w lr ∗ ∆lrλ

]

Long-range correlation energy Elrc

We have the following adiabatic connection formula:

E lrc =

∫ 1

0

dλ{

〈Ψlrλ|W

lr|Ψlrλ〉 − 〈ΦRSH|W

lr|ΦRSH〉}

=1

2

∫ 1

0

dλ Tr[

w lr ∗ P lrc,λ

]

and using the fluctuation-dissipation theorem

P lrc,λ = −

1

π

∫

∞

0

dω[

χlrλ(iω) − χRSH(iω)

]

+ ∆lrλ

where ∆lrλ comes from the variation of the density. So

E lrc = −

1

2π

∫ 1

0

dλ

∫

∞

0

dω Tr[

w lr ∗(

χlrλ(iω) − χRSH(iω)

)]

+1

2

∫ 1

0

dλ Tr[

w lr ∗ ∆lrλ

]

The long-range response function χlrλ(iω) is given by

χlrλ(iω)−1 = χlr

IP,λ(iω)−1 − f lrHxc,λ(iω)

Approximations for Elrc

Several approximations possible for E lrc :

TDHF approximation: f lrc,λ = 0 =⇒ RSH+TDHF method

Approximations for Elrc

Several approximations possible for E lrc :

TDHF approximation: f lrc,λ = 0 =⇒ RSH+TDHF method

MP2 approximation (2nd order in w lree) =⇒ RSH+MP2 method

Approximations for Elrc

Several approximations possible for E lrc :

TDHF approximation: f lrc,λ = 0 =⇒ RSH+TDHF method

MP2 approximation (2nd order in w lree) =⇒ RSH+MP2 method

Comparison:

RSH+TDHF is an extension of RSH+MP2

RSH+TDHF is expected to supersede RSH+MP2 for systemswith small HOMO-LUMO gap

Implementation of long-range TDHF

Orbital rotation Hessians:

(Aλ − Bλ)iajb = (ǫa − ǫi )δijδab + λ(

〈ij |w lree |ba〉 − 〈ia|w lr

ee |jb〉)

and

(Aλ + Bλ)iajb = (ǫa − ǫi )δijδab + 2λ〈ij |w lree |ab〉

− λ(

〈ij |w lree |ba〉 + 〈ia|w lr

ee |jb〉)

Implementation of long-range TDHF

Orbital rotation Hessians:

(Aλ − Bλ)iajb = (ǫa − ǫi )δijδab + λ(

〈ij |w lree |ba〉 − 〈ia|w lr

ee |jb〉)

and

(Aλ + Bλ)iajb = (ǫa − ǫi )δijδab + 2λ〈ij |w lree |ab〉

− λ(

〈ij |w lree |ba〉 + 〈ia|w lr

ee |jb〉)

Long-range TDHF second-order density matrix

P lrc,TDHF,λ = (Aλ − Bλ)1/2Mλ

−1/2(Aλ − Bλ)1/2 − 1

where Mλ = (Aλ − Bλ)1/2(Aλ + Bλ)(Aλ − Bλ)1/2

Implementation of long-range TDHF

Orbital rotation Hessians:

(Aλ − Bλ)iajb = (ǫa − ǫi )δijδab + λ(

〈ij |w lree |ba〉 − 〈ia|w lr

ee |jb〉)

and

(Aλ + Bλ)iajb = (ǫa − ǫi )δijδab + 2λ〈ij |w lree |ab〉

− λ(

〈ij |w lree |ba〉 + 〈ia|w lr

ee |jb〉)

Long-range TDHF second-order density matrix

P lrc,TDHF,λ = (Aλ − Bλ)1/2Mλ

−1/2(Aλ − Bλ)1/2 − 1

where Mλ = (Aλ − Bλ)1/2(Aλ + Bλ)(Aλ − Bλ)1/2

The TDHF long-range correlation energy is finally

E lrc,TDHF =

1

2

∫ 1

0

dλ∑

iajb

〈ij |w lree |ab〉

(

P lrc,TDHF,λ

)

iajb

1 Kohn-Sham DFT and ACFDT approaches

2 Range-separated multideterminant DFT

3 Short-range density functionals

4 Range-separated ACFDT method

5 Some results

Dependence on basis size: Ne2

Total energy (aug-cc-pVnZ basis, µ = 0.5, sr-PBE functional):

-258.2

-258.1

-258

-257.9

-257.8

-257.7

-257.6

-257.5

3 4 5 6

Tota

l en

ergy (

Hart

ree)

Size of one-particle basis (n in aug-cc-pVnZ)

ExactTDHF

RPARSH+TDHF

Ne2

=⇒ RSH+TDHF has a small basis dependence

Interaction energy curve of Ne2

Interaction energy (aug-cc-pV5Z basis, µ = 0.5, sr-PBE functional):

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

5 6 7 8 9 10

Inte

ract

ion

en

ergy (

mH

art

ree)

Interatomic distance (Bohr)

AccurateTDHF

RPARSH+TDHF

Ne2

Interaction energy curve of Be2

Interaction energy (cc-pV5Z basis, µ = 0.5, sr-PBE functional):

-4

-2

0

2

4

4 5 6 7 8 9 10 11

Inte

ract

ion

en

ergy (

mH

art

ree)

Interatomic distance (Bohr)

AccurateTDHF

RPARSH+TDHF

Be2

Conclusions and perspectives

Conclusions

RSH+TDHF method overcomes some problems of standardRPA

RSH+TDHF method seems well suited for van der Waalssystems

RSH+TDHF method has also problems (e.g., dissociation)

RSH+MP2 can be a cheaper alternative to RSH+TDHF

Perspectives

efficient implementation in quantum chemistry software

application to larger molecular systems (benzene dimer, ...)

application to solids

exploration of other variants of the method

Web page: www.lct.jussieu.fr/pagesperso/toulouse/